reserve TS for 1-sorted,
  K, Q for Subset of TS;
reserve TS for TopSpace,
  GX for TopStruct,
  x for set,
  P, Q for Subset of TS,
  K , L for Subset of TS,
  R, S for Subset of GX,
  T, W for Subset of GX;

theorem Th49:
  P is boundary & Q is boundary & Q is closed implies P \/ Q is boundary
proof
  assume that
A1: P is boundary and
A2: Q is boundary and
A3: Q is closed;
A4: Cl((P`) \ Q) = Cl(((P`) /\ Q`)) by SUBSET_1:13;
  P` is dense by A1;
  then
A5: [#] TS \ Q = Cl(P`) \ Q;
A6: Cl(P`) \ Cl Q c= Cl((P`) \ Q) by Th6;
  Q` is dense by A2;
  then
A7: Cl(Q`) = [#] TS;
  Cl(P`) \ Q = Cl(P`) \ Cl Q by A3,PRE_TOPC:22;
  then [#] TS \ Q c= Cl((P \/ Q)`) by A5,A6,A4,XBOOLE_1:53;
  then Cl(Q`) c= Cl(Cl((P \/ Q)`)) by PRE_TOPC:19;
  then [#] TS = Cl((P \/ Q)`) by A7;
  then (P \/ Q)` is dense;
  hence thesis;
end;
